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Questions tagged [linear-systems]

A linear system operates on inputs with only linear operators so the response to a complex input can be analysed as the sum of the response to a set of simpler inputs. This mathematical property makes the analysis of linear systems much simpler than non-linear systems where this summation or superposition does not hold. Linear systems are generally further classified as time invariant, meaning that there characteristics do not change over time.

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1answer
200 views

Conceptual Question from Signal Processing - Impulse Response and AR Coefficients

In continuation to the previous question Conceptual questions from signal processing I have a doubt which is: Consider an Autoregressive model (AR(2)): $$ y(t) = ay(t-1) + by(t-2) $$ and a FIR (...
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1answer
302 views

How can frequency dispersion in a vibrating string be modelled using an LTI filter?

According to JOS on stiff-string synthesis, stiff strings (like on a piano) introduce inharmonicity (i.e. the harmonics of the tone are not all in tune) due to different frequency components of the ...
2
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1answer
53 views

Find the length of the impulse response for the given output and input

Homework Question: Consider a signal $x[n]=\alpha e^{j \omega_{0} n}+\beta e^{j \omega_{1} n}+\gamma e^{j \omega_{2} n} .$ What is the length of impulse response $h[n]$ of a system (non-trivial) such ...
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1answer
719 views

Initial Rest Condition for LCCDE causal LTI systems

I am self studying Alan Opennheim's course Signals and Systems. I am a math major and have no background in EE. I understand that for a LCCDE system to be linear its auxiliary conditions must be 0. ...
2
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1answer
184 views

transient and steady-state response of first order system

Considering this general 1st order transfer function $$ H(z) = \frac{b_0 + b_1z^{-1}}{1-az^{-1}} $$ How to find (analytically) the transient and steady-state responses? With steady-state response I ...
2
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1answer
40 views

Reconstructed output mismatch for LTI system

I have a system with measured input (u) and output (y). I assume that this is an linear time-invariant (LTI) system and I want ...
2
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1answer
75 views

confusion in time invariance?

I am trying to learn shift invariance(time invariance) and i came across a function/system $$y(t)=3x(t)+2\cos(\pi t/3)$$ I am confused.If it was alone 3x,it should be definitely time invariant(i ...
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1answer
264 views

Causal LTI system having exponential input

I know that for an LTI system having complex exponential input, i.e, $x(t)=\exp(j w_o t)$ & $h(t) \to $ LTI System ; then, its output { $y(t) \} =M \exp(j w_o t + \phi)$ , where $M= |H(j w)|_{|w= ...
2
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1answer
809 views

Block Diagram for a difference equation

I have this little doubt regarding how to draw a block diagram representation of a difference equation. Let us implement $ y(n) = ay(n-3) + by(n-2)+cy(n-1) + x(n)$ in block diagram where $a, b, c$ are ...
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1answer
4k views

Determine if $ y[n] = ny[n-1] + x[n]$ is linear time invariant and BIBO stable

Check if the following system is linear time invariant and BIBO stable.. $$ y[n] = ny[n-1] + x[n] $$ for $n\ge 0$. We are also given that the system is at rest (i.e. $y[−1] = 0$). I know that to ...
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1answer
330 views

Eigenfunction property for LTI sinusoidal and the sinusoidal steady-state response

All LTI systems possess the eigenfunction property for complex exponential inputs. That is (restricting our attention to periodic complex exponentials), if $e^{j\omega_k t}$ is an input to the LTI ...
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1answer
84 views

Can linear transforms of vector space be seen as LTI systems? [duplicate]

Linear transforms of vector spaces has the linearity (i.e., homogeneity and superimposability), which is shared by LTI system. There are also shared concepts such as eigenvectors/eigenvalue... I am ...
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1answer
59 views

Notation of an LTI system consisting of LTI filters

I would like to find a reference for two notations of an LTI system consisting of LTI filters. In z-domain, the LTI system is given by $$ \mathbf{y}(z) = \mathbf{C}(z) \mathbf{s}(z) + \mathbf{D}(z) \...
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1answer
65 views

Is applying a linear operation to the input and output of a continuous linear operator OK?

The scientist and engineers guide to digital signal processing, Steven W. Smith, p. 134-135 states: [...] imagine a linear system receiving an input signal, $x[n]$. and generating an output signal, ...
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1answer
2k views

Using the linear operator to check for time invariance of a differential equation?

I have a differential equation $$\frac{d^2y(t)}{dt^2}+y(t) = \frac{dx(t)}{dt} + x^2(t)$$ and I need to see if this system with input $x(t)$ and output $y(t)$ is time invariant and linear. I tried to ...
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1answer
533 views

Difference equations applied to DSP

I've found almost nothing about difference equations on the internet. Can you please recommend me something like books or pdf online that handle deep this topic? I'm searching also some exercises. ...
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1answer
4k views

Is digital up and down sampling linear/causal/time-invariant?

So I am trying to determine whether the process of up sampling (zero padding) and downsampling are linear, causal and/or time-invariant. Based on some resources I found online, I am under the ...
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0answers
84 views

Find a LTI system such that $\mathcal{T}\{\frac{\sin t}{t}\} = \frac{\sin 2t}{t}$

Let $$x(t) = \frac{\sin t}{t} \qquad\text{and}\qquad y(t) = \frac{\sin 2t}{t}$$ Is it possible to find a LTI system such that $\mathcal{T}\{x(t)\} = y(t)$? If not, what's the reason for ...
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1answer
61 views

What are the properties of continuous-time improper systems?

I am trying to better understand the properties of improper systems $H(s) = \frac{b(s)}{a(s)}$, for which the order of the numerator $b(s)$ is greater than the order of the denominator $a(s)$ (in the ...
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1answer
59 views

Is the system $y[n]=x[n]+2=T\{x[n]\}$ an LTI-System?

Is the system $y[n]=x[n]+2=T\{x[n]\}$ an LTI-System? Linearity: $ \alpha \cdot T\{x_1[n] \} + \beta \cdot T\{x_2[n] \} = T\{\alpha \cdot x_1[n]\ + \beta \cdot x_2[n] \} \\\alpha \cdot (x_1[n]+2) + \...
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0answers
150 views

How to derivate in the frequency domain

I have two Time Domain functions, $f_1(t)$ and $f_2(t)$. I have both Fourier Transforms, $F_1(\omega)$ and $F_2(\omega)$. Functions $f_1$ and $f_2$ are not independent and, in fact, $f_1$ is also a ...
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1answer
52 views

Is my answer incomplete? Checking the stability of a system

Yesterday, during my exam, I had the following exercise: Given $$H(s) = \frac{1}{s^2+2s+4}$$ check if it's stable. which was supposed to be the hardest (since it was the last one). From my ...
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128 views

Check if I has right: Is this the Extended Kalman Filter

I have learn the Kalman-Buncy filter for the LQG controllers. I know that this is a signal processing forum and not robotics not math forum. But Extended Kalman Filters are daily discussed here. ...
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0answers
227 views

How does the ROC (Region of Convergence) related to a real world application?

In class, we are often given exercises to find the impulse response, output, and Z-transform of a system. In addition, we are often asked to define the Region of Convergence (ROC) depending on where ...
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2answers
67 views

Time-shift confusion

Say input-output of a system is defined as: $$ x[n] \longrightarrow x[nM] $$ then what will be the output of $x[n-1]$? will it be: \begin{align} x[n-1] \longrightarrow &x\left[(n-1)M\right] = ...
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0answers
54 views

Eigenvalues of correlation matrix which have the form of an harmonic function [duplicate]

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} &...
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0answers
436 views

How to evaluate performance of an ARMA, MA or AR model?

How to evaluate performance of a model after estimating ARMA/MA/AR parameters for any process x(n)? How to regenerate back a process after estimating average parameters? what kind of performance ...
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107 views

How to make equivalent system using the Noble Identity?

I am trying to get equivalent of this system using the Noble Identities: $$x[n] \rightarrow \boxed{H(z^L)} \rightarrow \boxed{\uparrow L} \rightarrow y[n]$$ my answer is $$x[n] \rightarrow \boxed{\...
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2answers
386 views

Time-invariant and Time-varying Systems

Determine whether the following system is time-invariant or not: $y(t)=x(t)\sin 10\pi t$ Solution: Given: $y(t)=x(t)\sin 10\pi t$ $y(t)=T[x(t)]=x(t)\sin 10\pi t$ The output due to input delayed ...
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3answers
2k views

Signals and systems : why do we study causal signals?

Till now I have read that causal signals are right sided and anti-causal, left sided. Why did we need to classify a signal with respect to its position? What is it's physical interpretation? ...
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3answers
1k views

LTI systems inverse of each other

So my question looks like this: Suppose that we have two LTI systems with impulse responses $$h_1(t) = \frac 12\delta(t-1)\quad\text{and}\quad h_2(t)=2\delta(t+1).$$ Determine whether these ...
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2answers
518 views

Alternative to BIBO stability of a system

In DSP textbooks a system is stable in the BIBO (Bounded-Input, Bounded Output) sense if and only if every bounded input sequence produces a bounded output sequence. After stating this definition ...
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1answer
97 views

What filter is better with -1/3 2/3 -1/3 or with -1/4 1/2 -1/4 coefficients?

Professor asked me what filter is better -1/3 2/3 -1/3 or -1/4 1/2 -1/4, the answer was second. But what those coefficients means? It was during discussion of linear filter but i'm not sure was it ...
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2answers
194 views

Why can convolution only be applied to compute the output of a linear filter?

We apparently cannot compute the output of a bilateral filter (BF) using convolution (with the image) because the BF is a non-linear filter. In general, why can convolution only be applied to compute ...
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2answers
120 views

Check whether a given equation is linear

$ a = (x, y) \in \mathbb{Z}^2 $ is given as a pixel. My equation in which $g$ is image(matrix) is defined as, $f(x, y) = 56g(x,y)+93g(x−1,y)+92g(x+1, y)−57g(x, y−1)+555g(x, y+1) $ How can we know ...
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1answer
649 views

Why is the total signal response response of zero input + zero state, $y(t) = y_0(t) + h(t)\star x(t)$, not an LTI?

For the zero input + zero state response in continuous time linearly time-invariant systems, why is the $y(t)$ equation not "technically" considered an LTI? I read this in a journal and there was no ...
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1answer
442 views

What is the physical interpretation of the dB scale on a bode plot and what is a negative dB?

I have no physical interpretation of the Bode plot. What does it mean for a bode plot to have negative dB over its entire duration on the log-scale frequency?
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2answers
59 views

Determining time-invariance of a system

I have a question about determining time-invariance of a linear system. We are given this system and we need to determine if it is time-invariant or not: $$y(t)=\int_{-t}^{\infty}x(-3\tau)d\tau$$ ...
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3answers
48 views

System classification: unit-time delay

I'm reading a book on linear systems and I can't understand why the unit-time delay is a distributed system. This is the example given in the book: I understand that the initial state of the system ...
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1answer
124 views

is y[n] = x[n] + n time invariant?

My steps were as follows: $\ x_2[n] = x[n-k] $ $\ y[n-k] = x[n-k] + (n-k) $ and $\ y_2[n] = x_2[n] + n = x[n-k]+(n-k)$ Does this mean that it is indeed time invariant?
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1answer
1k views

Fourier transform of function division in time domain

$F_1(\omega)$ is the Fourier Transform of $f_1(t)$. $F_2(\omega)$ is the Fourier TRansform of $f_2(t)$. Can I obtain the Fourier Transform ($F_3(\omega)$) of $$ f_3(t) = \frac{f_1(t)}{f_2(t)} $$ ...
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2answers
231 views

should this be viewed as an impulse response or step response

I'm trying to teach myself the relation between simple discrete ODE's and the impulse response-step response concept. Getting back to the question: I don't expect anyone to read the whole thing but I'...
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1answer
429 views

BIBO stability of $1/x(t)$

I realize this is quite basic stuff but I am having trouble with the following question: Determine whether the following system with input $x(t)$ and output $y(t)$ is BIBO stable: $$y(t) = \frac{1}{...
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1answer
179 views

$\mathcal Z$-transform ROC

Let's say I have a $\mathcal Z$-transform that represents some transfer function and its has some ROC. My question is how do I know if this system is causal? I know that if the ROC contains the ...
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2answers
385 views

Why call it discrete time signals as difference equations?

There are lots of representations of discrete time signals. You represent it in block diagram or as difference equations. We call it block diagrams as there are elementary structures in the diagrams. ...
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2answers
304 views

Output of a discrete-time LTI system different form than input?

This question is related to this one. I'm going through old exams for a 2nd year systems and transforms course, and came across this question. I'm posting this question just in case my other question ...
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2answers
2k views

How to prove that frequency mixer is a nonlinear system?

I know that frequency mixer $y(t)=x(t)cos(\omega_ct)$ is a nonlinear system since its output has frequency components that are not present in the input. To prove its nonlinearity, it shuldd be shown ...
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2answers
109 views

Is this HW question even valid? (linear systems)

The question is: Are the following systems linear time invariant? $y(t)=x(t)$, with $x(t)=u(t)$ (unit step function) $y[n]=x[n]$, with $x[n]=\delta[n]$ (Dirac delta function) The reason I am asking ...
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2answers
42 views

Stability of passive LTI system

Is a passive LTI system will always be a stable(bibo and lyapunov) system? Or in other words Passivity of a LTI system implies stability? If above statements are true then there must be some ...
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1answer
34 views

Transients in marginally stable system

I know that if a system is asymptotically stable then after infinite time its output contains only characteristic of input signal and it is also known as steady state and there are no transient ...

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